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Full text of "Solution of the telegrapher's equation with boundary conditions on only one characteristic"

U. S. Department of Commerce 
National Bureau of Standards 



Research Paper RP2059 
Volume 44, January 1950 



Part of the Journal of Research of the National Bureau of Standards 



Solution of the Telegraphers Equation With Boundary 
Conditions on Only One Characteristic ] 

By George E. Forsythe 

Forecasting a certain idealized horizontal, autobarotropic, nonviscous, nondiverging 
atmospheric flow considered by Rossby leads to an unusual boundary-value problem for 
the telegrapher's equation, involving boundary values on only one characteristic. It is 
shown how to find unique solutions periodic in the longitude; these are represented in terms 
of a Green's function, A procedure for computing the Green's function is set down and is 
shown to be optimal in a restricted sense. The Green's function is tabulated for 72 longi- 
tudes and 14 time-values. An alternative solution by a difference equation is mentioned. 



I. Introduction 

In one treatment of planetary atmospheric flow 
as horizontal, autobarotropic, nonviscous, and 
nondiverging in a plane, Rossby [9] 2 considered 
the idealized case of a constant west-wind com- 
ponent, U, and a south-wind component, v, depend- 
ent on the west-to-east distance coordinate,^, and 
time, r, but independent of the south-to-north 
distance coordinate, /x. It was shown in [5] that this 
v satisfies the telegrapher's equation (eq 3) below, 
where x=\[/—Ut, f=4 fir. The parameter £= 
2Q cos <p (d<p /did) is here considered constant; 12 is the 
angular speed of the earth's rotation, and <p is 
latitude. For this simple atmospheric model, the 
meteorological forecast problem is one of deter- 
mining v(x, t) for future times t, given only v(x f 0). 
But on a plane the specification of v(x, 0) is not 
sufficient to determine v(x, t) for many £>0, be- 
cause the line t—0 is a characteristic of eq 3 (see 
p. 254 of [11]). Having its initial conditions on 
only one characteristic is an unusual feature of 
the present problem that does not seem to have 
arisen in other physical problems known to the 
author to lead to the telegrapher's equation. 

The author shows that the forecast problem has 
a unique solution when it is assumed that the world 
is round, that is, when the solution is assumed to 

1 This paper was written at the Institute for Numerical Analysis of the 
National Bureau of Standards with the financial support of the Oflice of 
Naval Research of the U. S. Navy Department. 

2 Figures in brackets indicate the literature references at the end of this 
paper. 



be periodic in x. The problem is stated in section 
II and solved in section III. In section IV the 
solution is represented in terms of a Green's func- 
tion. In section V a procedure is outlined for 
computing the Green's function by improving the 
convergence of its Fourier series. In section VI 
certain auxiliary polynomials, a k (x), used in 
seel ion V are discussed and related to the Bernoulli 
polynomials. In section VII are reported without 
proof a few results on the approximate solution of 
the problem by a difference equation, taken from 
[(>]. In section VIII is given a table of values of 
the Green's function, as computed in the Compu- 
tation Unit of the Institute for Numerical Analysis. 
The present author first reported this work in 
[7]. Independently of the research reported here, 
Charney, Eliassen, and Hunt of the Institute for 
Advanced Study considered the telegrapher's 
equation while investigating numerical weather 
prediction in general. Their research was reported 
in [1] and is written up in [2]. The work of these 
men includes much of what is reported here, and 
much more. 

II. Statement of the Problem 

Let C be the circumference of a unit circle; let 
us adopt an angle coordinate x for C\ — tt<^x<tt. 
Let / be the set of time-instants t: 0<t<^ °° . Let 
R be the closed two-dimensional region consisting 
of all points (x, t) with x in C and t in /. Let f(x) 
be a real-valued function that satisfies the follow- 



Solution of the Telegrapher's Equation 



89 



ing hypotheses, but which is otherwise arbitrary: 
Hi. f(x) is sectionally smooth 3 on C. Moreover, 



f(x)=i[f(x+0)+f(x-0)], (all x). 
H 2 : fix) has the average value zero: 

f(r)dx=0. 



/: 



(i) 



(2) 



The problem is to find a real-valued function, 
v(x, t), defined everywhere on i?,with the following 
four properties: 

Pi'. v t exists* and is continuous throughout R. 

P 2 : v x and v tx =c)Vt/c>x exist and are continuous 
everywhere in R except, at most, for a finite number 
of values of 5 x. 

Pz'. Whenever v xt is defined, the following hyper- 
bolic partial differential equation (the telegrapher's 
equation) is satisfied: 

v xt +lv=0. (3) 

P 4 : For t=0, v(x, t) reduces tof(x): 

»(3,0)=/(3). 

III. Solution of the Problem, Uniqueness 

One gets a formal solution by separation of 
variables 'and use of Fourier series. Assume a 
solution of eq 3 of form v(x, t) =X(x) T{t) . Then 
v x .=X'(x)T'(t), and eq 3 takes the form 



X'V) T'{t) 
X(x) ' T(t) - 



(4) 



The two factors in eq 4 must themselves be 
constant: 

X'(x) , 



X(x) 

T'(t) _ 
T(t) 



_1_ 
"4\* 



(5) 
(6) 



A solution of eq 5 for — oo<^x<oo is X(x)=e Kx . 
For x on the circle (7, however, one must have 
X(—Tr)=X(ir), or e"* x =€* x . Taking logarithms, 
oneseesthat — 7rX=7rX+2n^(?i=0, ±1, ±2, . . .). 
Hence X=m(7i=0, ±1, ±2, . . .). Since the value 



3 That is, both f(x) and/'(z) are continuous in C except for a finite number 
of jump discontinuities. 

4 The subscripts denote partial derivatives. 

» It is shown on pp. 55 to 57 of [3] that our conditions Pi and P 2 imply the 
following: r xt exists and equals Vt x for all (x, t) such that x is not one of the 
excepted values in P 2 . 



X=0 is incompatible with eq 4, there remain the 
following fundamental solutions of eq 5: 



X n (x)-- 



(n=±l,±2, ±3, 



.)■ 



Corresponding to X n (x), the solution T n (t) of eq 6 
for \=ni is T n (t) = exp (it/4n) . Hence for n= ± 1 , 
±2, . . . the functions X n (x)T n (f)=exp [i(nx+ 
t/4:n)] have properties P u P 2 , and P 3 . By taking 
linear combinations of the functions X n T n and 
X_ n T_ nj one obtains the equivalent pair of func- 
tions cos (nx-\-t/4:n) and sin (nx-\-t/4ri). Both of 
the latter functions have properties P l9 P 2 , and P 3 . 
In order to obtain a solution with enough de- 
grees of freedom to satisfy P 4 , consider the series, 



v(x,f)' 



-§ [ a » cos ( nx+ L) +b " sin ( w+ ^)]' 



(7) 



where a n , b n are undetermined constants. We 
postpone a discussion of the convergence of the 
series (eq 7) for t^O and consider it for t=0, where 
v(x } 0) is supposed to equal f(x): 



v(x,0) ^^Li(a n cos nx-\-b n smnx). 

71 = 1 



(8) 



If the series in eq 8 actually does converge tof(x) 
for all x, it is shown on p. 274 of [12] that the co- 
efficients, a n , b n , must be the Fourier coefficients of 
/. Conversely, by p. 25 of [12], the hypothesis 
Hi is sufficient to insure that the Fourier series of 
/ actually converges to f(x) ; it even converges uni- 
formly for x in any interval bounded away from 
a discontinuity of /. Moreover, the hypothesis 
H 2 implies that in the Fourier series a =0. We 
henceforth stipulate that the series (eq 8) is the 
Fourier series of/. It is important to note 6 that 
Hi implies that a n and b n are 0(l/n); that is, there 
exists a constant M <C °° such that 



\na n \<M, \nb n \<M (all n). 
1. Proof of Convergence 



(9) 



There remains only a proof that the series in 
eq 7 actually does converge to a function v(x, t) 
with the required properties, Pi,P 2 , P 3 , P 4 . It will 
be useful to have the following representations of 
cos (tjAn) and sin (tjkn). They are proved by 
Taylor's formula and hold for all values of t/An: 

« See p. 18 of [12]. 



90 



Journal of Research 



cos-j— =1" 

An 



* n t 2 
32n 2 



where|a„|=a^j|<l; 



. t U 
An An 



where | /S,| =|^(^) ! <1; 



sin 4n 4n 384n 3 ' 



(10) 



(11) 



(12) 



where \y n \ =|t(^)| <1. 

Using eq 10 and 11, one sees that for any fixed t, 
S «* cos(nx+^J+b n sm(nx+ 4 - ) I 

S ( a n cos "£+&* sin nx) cos vz + 
(&„ cos nx— a„ sin ma;) sin ~. = 

oo 

S ( a « cos nx+bn sin m) — 

71 = 1 

t^ (b n a n . \ 
-7> j p n ( — cos nx sin nx )= 

4^ ^ n \n n J 



(13) 



Representation as the sum of three series is per- 
mitted because each of the series 2 , S x , S 2 
converges. 2 converges for all x because it is 
the Fourier series of f ; its convergence is uniform 
in any interval bounded away from a discontinuity 
of f(x). Fix any positive number t l9 and restrict 
the consideration to t'& such that 0<t<ti. Since 
a n and b n are 0(l/n) 9 2i and 2 2 are convergent 
uniformly in x and 2. For example, 2 2 is domi- 
nated by (U/A) ^,(\b n \/n-\-\a n \/ri), a series con- 
vergent like 2(lM 2 )- The series of eq 7 is thus 
convergent for all x, t and defines by its limit a 
function v(x, t): 

v(x, 0=S &n cos (^+4^)+&n sin ( nx+~ ) • 

(14) 

Moreover, the series (eq 14) converges uniformly 
for x, t such that x is bounded away from a jump 
of/(x). 



Since Si and 2 2 converge uniformly, they con- 
verge to continuous functions of x and t. Thus 
the only discontinuities of v{x, t) are those from 
S , that is, those of j(x). This is the property of 
hyperbolic differential equations that discon- 
tinuities in their solutions are propagated along 
characteristics. Let the set of discontinuities 
of J(x) be denoted by E. 

For all x, t one may obtain v t by termwise dif- 
ferentiation of eq 14, because by eq 9 the resulting 
series is absolutely and uniformly convergent in 
both x and t: 

*(*, =*£ g cos (W ±)-<% sin (««+£)} 

(15) 

Moreover, v t is continuous for all x, t, so that Pi 
holds. Now one may not obtain v x by termwise 
differentiation of eq 14, because the resulting 
series will generally not converge. However, 
v x does exist and is a continuous function of x 
and t for all t and for all x not in E. To see this, 
one uses eq 10 and 12 to carry the Taylor formula 
(eq 13) to one higher power of t. It is found that 



)(Xj t) =f(x) +t ^ ( " 



'K 

n 



On 

n 



)- 



t 2 
32 






b n 



cos nx-\ — 2 sni 7i ^ 



■)- 



3kS 7 »(«» COS7w: -S 8in7la; } (16) 

By termwise differentiation of eq 16, it is found 
that 



4- OO 

#*(#, t) =j r (jJ-tS (a B cos nx+ 6 n sin nx) — 

4 71 = 1 

t 2 ™ /b n a n . \ . 

™ >J a w ( — cos nx sin nx ) + 

32 ^i n \n n ) 



384 



s*& 



cos tmH — 2 sin nx 



/'(x)-i#)+2 3 +2, 

Restricting attention to t with ^ ^ /i< °° , one 
sees that the expressions 2 3 and S 4 are uniformly 
convergent with respect to x and t. The series 
leading to j(x) is uniformly convergent for x 
in any interval bounded away from a discontinuity 



Solution of the Telegrapher's Equation 



91 



of /(a?). Hence, for x not in E, termwise differ- 
entiation leads to the correct value of v x . More- 
over, v x is continuous in x and t whenever a; is a 
point of continuity of f(x). To get v tx one may 
differentiate eq 15 termwise with respect to x: 

v tx (x,t) = 

"*S[ a - co *( nx+ in) +bn sin (^+^)} < 17 ) 

As remarked after eq 14, the series in eq 17 is 
uniformly convergent for x, t such that x is bounded 
away from the discontinuities of f(x). Hence 
v tx is continuous in x and t for all x, t except for 
x in E. Since v t , v x , v tx are all continuous, v xt 
exists and equals v tx except on the lines corre- 
sponding to the discontinuities of /(a?). This 
shows that v(x, t) has property P 2 . The eq 17 
and 14 show that v satisfies the telegrapher's 
equation (eq 3). Finally, property P 4 was taken 
care of by the selection of {a ny b n }. Thus the 
problem is solved completely. 

2. Proof of Uniqueness 

It will be shown that v(x } t) is the only function 
that solves the above problem. Suppose that 
Vi(x,i) were a second solution. Then the differ- 
ence, w(x,t)=v—Vi, satisfies the same problem with 
f(x) =0. For each x u by property Pi, w{x u t) and 
w t (xij t) are continuous functions of t for 0^t<C °°, 
while w(x, 0) = and w x {x, 0)^=0 are, of course, 
continuous functions of x. " ' For each t, let the 
value of w(x, t) be extended as a periodic function 
of x to all x in the interval [— 2ir, 2ir]. Now the 
values w{— ir, t) and w(x, 0) are given on two charac- 
teristics of eq 3. By pp. 21 to 22 of [10] they are 
therefore sufficient to determine w(x, t) uniquely 
for all x, t. On the other hand, the values w(t, t) 
and w(x, 0) are also sufficient to determine w(x,t) 
for all x, t. Since w(—tt, t) =w (vr, i) and w(x, 0) = 
w(— x, 0) = 0, it is seen by symmetry that w(x, t) = 
w(—x,t). Now since the values of x lie on a circle, 
there is nothing exceptional about the line£=7r. 
The above argument will also show that, for each 
value of Xi, w{x\-\-x } t) = w(xi—x, t). It follows that 
for each fixed t, w(x, t) = constant, whence w(x, t) = 
hit). Byeq3, —iw=w tx z= (d/dx)h / (t) = 0. Hence, 
the constant value of w(x, t) must be everywhere 
zero. Then v=v h and the solution v(x, t) given by 
eq 14 is unique. 



The results of section III may be summarized 
in the following theorem, phrased in the notation 
of section II. 

Theorem 1. If the real-valued junction f(x) 
defined on C satisfies hypotheses Hi and H 2 , then 
there exists a unique function, v(x, t), defined on 
R and possessing properties P u P 2 , Pz, and P^ 
If eq 8 is the Fourier series of f(x), then v(x, t) is 
defined explicitly by eq 1/+. 

It is of mathematical interest 7 to note that 
Theorem 1 can be extended to general functions 
f(x) of bounded variation. That is, one may re- 
place Hi by the weaker hypothesis 

Hi: f(x) is of bounded variation on C. Moreover* 

f(x)=i[f(x+0)+f(x-0)], (all*). 

The solution v(x, t) is required to have property 
P 2 , instead of P 2 : 

P 2 : v x exists in R except for x in a set E (EcC) 
of Lebesgue measure zero; for all t and for all x not 
in E, v tx exists and is a continuous function of x and d t. 

The extension of Theorem 1 is stated as follows: 

Theorem 2. If the real-valued function f(x) 
defined on C satisfies hypotheses Hi and H' 2 , then 
there exists a unique function v(x, t) defined on R 
and possessing properties P u P' 2 , P 3 , and P 4 . If 
eq 8 is the Fourier series of f(x), then v(x, t) is 
defined explicitly by eq H. 

The convergence proof of section III, 2 requires 
only slight modification to serve as a proof of 
Theorem 2. For an arbitrary function f(x) of 
bounded variation, there need be no interval of 
continuity; one may therefore not expect the 
series (eq 14) to converge uniformly in any 
interval. The termwise differentiations of section 

III, 2 can, however, be justified for almost all x by 
the fact that the resulting series are Fourier series. 

IV. Representation by a Green's Function 

The formula (eq 14) for the solution of the 
problem of section II is directly adapted to 
numerical computation only when the Fourier 
coefficients a n) b n converge rapidly to zero. But 
some of the most important cases in meteorology 
are where f{x) has discontinuities (see footnote 7). 

7 This extension seems to have no meteorological interest. However, it is 
of much importance in meteorology to deal with functions f{x) with some dis- 
continuities; such discontinuities occur at fronts between air masses. 

8 Same aseq 1. 

9 It follows from Pi and P' 2 that v xt exists and equals v tx for all t and for all 
x not in E; see pp. 55 to 57 of [3]. 



92 



Journal of Research 



With such an / the Fourier coefficients are, 
roughly speaking, of the order 0(1 /n), and for 
those / the convergence of eq 14 is hopelessly 
slow. 

It is possible, however, to improve the con- 
vergence of eq 14 to such a degree that computa- 
tion of v(x y t) is reasonably possible. The pro- 
cedure will be illustrated in section V for one 
particular choice of f(x) : 

v sin nx 



f(x)=<r (x)-- 



71-1 



(0<x<x) 

(x=0) 

■iU+x) (-w<x<0), 

(18) 

It may be shown by direct computation that the 
Fourier series of oo(x) is the series of eq 18. It 
then follows that the series converges to a {) (x) for 
all ./'. The reason for choosing a (x) is two-fold: 
(a) it is of meteorological interest to see how a 
simple discontinuity in v(x, 0) is propagated, as t 
increases; (b) for any f(x) that is sectionally 
smooth, it is possible to represent the correspond- 
ing v(x, t) in terms of the solution for the special 
initial condition v(x, Q)=a (x). 

The present section is devoted to proving the 
property (b). Suppose, therefore, that G(x,t) is 
the solution of the problem of section II with the 
initial condition <r (x)) then G(x, 0) = a (x). Let 
a sectionally smooth function f(x) be given that 
satisfies eq 1 and 2. Let /(#) have the jump J k = 
f(x k +0)—f(x k —0) at the point x k (k = l, 2, . . . ,K). 
Let Co Or) be continued periodically for x in [— 2ir, 
2w\. Then (J k /ir)a (x—x k ) also has the jump J k 
at the point x k . Now 

1 K 

£0*0 =f( x ) S Jk<r (x—x k ) 

7Tfc=i 

is a continuous function, since all the jumps have 
been removed. Moreover, £(x) and the functions 
(J k /ir)<r (x—x k ) all satisfy eq 1 and 2. There is, 
therefore, a unique solution to the problem of 
section II for each of these functions. For the 
function (J k /Tr)<r (x—x k ) the solution is (Jjc/tt) 
G(x—x k , t). It will be shown below that the solu- 
tion y(x, t) corresponding to the initial values £(x) 



is given by 

1 f * 
y(x,t)=- G(x-u,t)£'(u)du. (19) 

Since the problem of section II is linear in the 
initial condition /(#) , and since 

/(*) =*(*)+- S eWs-z*), (20) 

it follows that 

1 K I r^ 

v ( Xj t)=-^J k G(x—x k ,t)+- G(x-u,t)Z'(u)du. 

(21) 

The formula (eq 21) is the desired representation 
of v(x, t) in terms of the solution G(x, t) to the single 
problem where f(x) = o- (a;) . The nature of eq 21 
indicates that G(x,t) may be called the Green 7 s 
function of the problem of section II. The repre- 
sentation (eq 21) is not only of theoretical impor- 
tance, but it can also be used for approximating 
the solutions for general boundary values, fix), 
once the Green's function is tabulated. The prac- 
tical problem then becomes one of approximating 
the integral in eq 21 by some numerical process. 
This latter problem is not treated here. 

It remains to prove eq 19. First, it may be 
observed that, for each x, since <r and £ are 
periodic, 

a (x—u)£' (u)du=- a (;x—u)d£(u) = 

Wj-TT TTj-TT 

i r *■ i r x ~° , 

£('u)d u <T (x — u) =- £(u)a (x—u)d 

TTJ-tt TTJ-tt 



II 



i C w 

+ t(x)+- £(u)a (x—u)du= 
irjx+o 

1 f* 

£ (x) - ^ J _J (u) du=£ (x) . 



(22) 



In eq 22 we have used two Riemann-Stieltjes in- 
tegrals. The last step is true because £(x) 
satisfies eq 2. By eq 13, 

G(x, f)=a Q (x) +J S § cos nx-~ J2 °^ sin nx. 
Hence 



if* i f 7r t f 71 " r^% "l 

G(x—u,t)£'(u)du = ~ <r (x—u)£'(u)du-{--7— z_, ~| (cos nx cos nu-{- sin nx sin nu) \£'(u)du— 

7T J-7T KJ-Tr 47T J-ttLw=1 n J 

t 2 f* r " a n/ . . ."!,.,. . . (23) 

^T" > i -s (sm nx cos nu— cos nx sin nu) \£(u)du. 

6ZT J-7T L W=l n J 



Solution of the Telegrapher's Equation 



93 



Since £(u) is sectionally smooth, \£'(u)\ is bounded. 
Hence the series in eq 23 remain uniformly con- 
vergent when multiplied by %'(u) and may be inte- 
grated termwise. We note that 

- cos nu%'(u)du=- sin nu%(u)du= nb n , 

7T J-7T 7T J-7T 

- $mnu£'(u)du= : cos nu£(u)du= — n a n , 

(24) 

where a n and b n are the Fourier coefficients of f. 

In view of eq 22 and 24, the termwise integra- 
tion in eq 23 yields 

- C G(x-u,t)Z'(u)du=ttx) + 

TTJ-TT 

t °° B 

j^Lj— (b n cos nx—a n sin nx) — 

4 n= i n 



OL n 



™ 2 d? («n cos w«+ ft« sin nx) . 

oz w= ] n 



(25) 



But, by eq 13 and 14, the right-hand side of eq 25 
is y(x,f), the solution corresponding to the initial 
values £(#). This completes the proof of eq 19. 

The representations (eq 20 and 21) assume a 
more symmetric and unified form when the Le- 
besgue-Stieltjes integral is used. It can be shown 
that 

/(*)=-( <T (x-u)df(u), (26) 

and that 

»(M)=-f G(x-u,t)df(u), (27) 

TTJ C 

where eq 26 and 27 include Lebesgue-Stieltjes in- 
tegrals over the circle C. Whenever f(x) has a 
discontinuity (say at x{) , the integral in eq 26 fails 
to converge as a Kiemann-Stieltjes integral for 
x=Xi, because the. functions o- (xi—u) and f(u) 
both have a discontinuity for u=0. The same 
holds for eq 27. The integrals (eq 26 and 27) are 
convergent for all x as Lebesgue-Stieltjes integrals. 
Moreover, the formula (eq 27) yields the solution of 
the problem when f(x) is an arbitrary function of 
bounded variation; the above proof of eq 19 can 
be modified to serve as a proof of eq 27. 



V. Computation of the Green's Function 

For the purpose of using the representation (eq 
21) and for its own meteorological interest, it was 
desired to compute the Green's function G(x, t). 
A tabulation to three decimal places, accurate to 
approximately 0.001, appeared sufficiently accu- 
rate. An x interval of 5 degrees of longitude 
(7r/36 radian) is convenient in meteorology. It 
was decided to compute G(x, t) for various times 
t up to 96 hours at latitudes <f> from32°19 / to 10 
55°. Since the length unit is here the radian of 
longitude, the expression d<t>/djjL in section I takes 
the value cos <j>. Then 40=812 cos 2 <£, where 
ft=7.292X10" 5 radian/second. Now 24 hours 
corresponds to r =86,400 seconds, or to t= 50.40 
cos 2 <£. At latitude 32°19 / , cos 2 0=0.714, whence 
£=36 at 24 hours. The largest value of t for 
which G(x, t) was computed corresponds to 96 
hours at latitude 32°19'; it is £=144. 

Let 2=//4, for convenience. By eq 14 



G(x, z) =X1 - sin ( nx-\ — )• 



(28) 



In summary, a method is required to compute 
G(x, z) to an accuracy of approximately ±0.001 
for 11 x= — 7r(7r/36)7r and for various positive values 
of z up to 36. The present section will present 
one such procedure, an application of a 
method for improving the convergence of certain 
Fourier series, given on pp. 84 to 88 of [10]. The 
'procedure presented below is not an exact description 
of the methods actually used in making the table of 
section VIII. It is assumed in section V that 
computing machinery is available capable of 
dealing with numbers of 10 decimal digits, but 
no more than 10. 

Of the tolerable error 0.001, the amount 0.0005 
must be reserved for round-off in the final tabula- 
tion to three decimal places. Suppose that 0.0004 
is allowed for truncation errors, 12 and 0.0001 for 
computing errors resulting from round-off s during 
the calculation with 10-digit numbers. To have a 
truncation error as low as 0.0004 from use of a 
partial sum of eq 28 would require about 23,000 

!« The limit <£=32°19' arose unintentionally. 

11 The notation x=a(8)b means x=a, a-\-8, a+25, a+38, . . . , 6—5, b. 

12 Truncation errors are errors that result from use of approximate mathe- 
matical formulas, e. g., use of partial sums of infinite series. 



94 



Journal of Research 



terms for x=7r/36 and #=357r/36. The con- 
vergence must obviously be improved. 

1. Representation by Truncated Double Sum 

We write 

00 1 z 

G(x, z) = ^Pi — sin nx cos -+ 

n — l Jl 71 

00 I 2 

Xj - cos nz sin -=Si+S 2 ; (29) 

n = l ^ ^ 

for simplicity we consider only the first sum Si in 
eq 29. It can be shown that the term S 2 behaves 
similarly throughout the analysis. Expanding 
cos (z/ri) in its Maclaurin series, one has 

v-^ 1 • ^ ( -l) r z 2r _^ ^ (-l) r z 2r sin nx 
Zl ~ir{ n sm nx fa (2r) !n» -£j fa (2r) to 2 '* 1 

(30) 

One type of truncation of eq 30 consists in 
omitting all terms for n>N-\-l, r>R. Let the 
error caused by this truncation be called ei. We 
shall estimate e x for 0<z<Z: 



«i = 



n = A 7 +l r=/t! 



(— l) r ^ 2r sin ra 



(2r)!^ 2r+1 

'! Jn x 
« /eZV 



oo V2r oo 1 oo 72r 



= s 



L'r i 1 



£4 (2r) (2r)! ^ 4 V* 

The last step above uses the Stirling expression 
for the factorial function, which, according t o p. 
74 of [8], is a one-sided estimate: s\ > s s e~ s ^2irs. 
Continuing, one finds for <?Z/2i?iV<l that 

i i^l f eZV R 1 ^( eZ V r 
{eil< -±^-\2RNj B*>*fa\2RNj 



4V7ri? 3/2 

The first estimate in eq 31 seems crude, but it 
does not affect the values of N or R very much. 
Thus F(A T ,R) is an upper bound for the trunca- 
tion error |ei| introduced by omitting terms of 
type n>N+l, r>R in S lt 

One can reasonably tolerate a truncation error 
|ei| of 5X10 -5 . The corresponding admissible 



values of N and R come from setting F(N,R) = 
5X10 -5 for Z=36. The following pairs of values 
of N,R were obtained from eq 31 by a numerical 
calculation followed by a round-off of N to an 
integral value: 



N=2Q 
18 
13 
11 



iJ=4 



N=9 

7 
4 



R=S 



10 



15 



20 



(32) 



The selection of the most suitable pair of values 
N,R from eq 32 will be postponed until we have 
discussed the summation of the remaining terms 
of eq 30. 

2. Computation of the Double Sum 

The terms of Si for n=l, 2, . . . , N and all r 
may be left in the form 



S 1 =>,( - cos- )si 
»=I \n n/ 



sm nx, 



(33) 



and may be computed from this formula. The 
terms for n>N+l and r=0, 1, . . . , R—l may be 
written in the form 



where 



S '~~§(2r)! a * )(x) ' 



4fr» = (-iy £^r 



(34) 



(35) 



Once g^ (x) has been tabulated for the one value 
of N to be selected below, Sg may be computed 
directly from eq 34. Two methods are needed to 
get a^ } (x), as the calculating machinery is as- 
sumed to be limited to ten decimal digits. 
The first method is to use the identity 

c%\x) = c 2T {x)-{-iyJt S -^g' (36) 



where 



«2r(x)=^(x) = (-iy± S -^- (37) 



In section VI it will be shown that for 0<x<7r, 
g 2t {x) is essentially a Bernoulli polynomial in the 
variable x/2w. Hence <r 2 r(x) can readily be corn- 



Solution of the Telegrapher's Equation 



862453—50- 



95 



puted or even interpolated from existing tables 
like [4]. One can then get cr^ix) by carrying 
out the subtraction indicated in eq 36. The terms 
a 2r (x) and (sin x)/l of eq 36 are approximately 
unity; with ten-digit calculating machinery these 
terms may be carried to ten decimal places. Hence 
0-2^ (x)j calculated from eq 36, will be good to 
ten decimal places. In order to get the Green's 
function to an accuracy of 0.001, it is necessary 
that all terms in the sums of eq 33 and 34 be sub- 
stantially correct to five decimal places; hence the 
terms z 2r o^ (x) I (2r)\ in eq 34 must be given to five 
decimal places. It follows that z 2r j(2r)\ must be 
less than 10 5 ; for 2=36 this means that 2r<4. 
Thus the first method of computing a^\x) is 
adequate when 2r=0, 2, 4. 

For 2r > 4, another way of getting o-^° (x) is 
needed. For these larger r, the convergence of 
the infinite series (eq 35) is good, and we may 
write 

<^>(^(-ir 2 S 4|?- (38) 

n=N+l n 

The truncation error e 2 in eq 38 may be shown 
to be at its maximum when z=7r/36 and when 
sin (N 2T x) =0. In this case the error is not greater 
than the sum of the first 36 omitted terms, which 
is estimated by 



^ 36 s in (rnr/36) . 1 



w=AT 2r +l 






mr ^ 72 



^iV 2 2 ; +1 tA ° ll± 3^7rN 2 2 ; +1 



In order to keep € 2 z 2r /(2r)\ numerically less than 
5X10 -5 for ^=36, it is sufficient that 

KfJ +1 (^ =5Xl(r - (39) 

Solution of eq 39 gives the following points of 
truncation of eq 38: 



iV 4 =126, N 5 =78, iV 6 =54, JV 7 =40, 
^8=32,^=26,^0=22,.... 



(40) 



(The values of N k for odd subscripts k are appro- 
priate to a parallel analysis of the sum S 2 of eq 29, 
but not to the present analysis of S x .) If ci N) (x), 
<rs N) (x), . . . , oiftl.2 (x) are estimated by eq 38, with 
the values N 2r taken from eq 40, the individual 
summands of eq 34 will each have a truncation 
error not exceeding 5X10 -5 . 



3. Operational Analysis, Selection of N and R 

We now estimate the labor involved in comput- 
ing the Si of eq 29, in order to select that pair of 
values of N and R from eq 32 that makes the 
computational work a minimum. The resulting 
computing procedure will be optimal in a limited 
sense — i. e., optimal among the one-parameter 
family of truncations considered in section V, 1. 
Although the resulting procedure will be perfectly 
feasible for computation — indeed, it differs only 
moderately from the procedure actually used to 
get the tables of section VIII — it cannot be said 
to be optimal among all procedures for computing 
the Si of eq 29. For it has been based on a certain 
type of truncation of a certain double series (eq 
30), and on a predetermined assignment of 
truncation errors to several subcalculations (eq 
38). Given only the nature of the computational 
machinery, to describe an absolutely optimal 
procedure of getting Si would seem quite beyond 
the present powers of analysis. 

It is customary and quite realistic to estimate 
the cost of a computation by the number of 
multiplications required. 13 We shall consider 
the multiplications required to get Si for one 
value of z and for one value of x. In eq 33 one 
may ignore the desk computation required to get 
(l/ri)cos(z/n) and sinnx; there are then essentially 
N multiplications involved in eq 33. In using 
eq 36, one may ignore the work of getting <j 2r (x), 
which is chargeable to basic table development, 
and count SN multiplications needed in all to 
get a<$>(z), ff ( ?W, <r { T(x). To get <r™(x) from 
eq 38, in view of eq 40, requires 54— iV multipli- 



cations. To get a { %\ <j { \ 



from eq 40 



requires in all approximately (22—4) (30— iV) multi- 
plications, where 30 is a rough average of the 
higher values of N 2r in eq 40. To put Sg together 
by use of eq 34 involves R multiplications. 

Summarizing, we find that getting Si by the 
outlined procedure requires for each value of z 
and each value of x the number of multiplications 



W 1 (N,R) = -m + 7N+3lR-RN= 
151- (7-22) (31 -N). 



(41) 



Minimizing Wi(N, R) over the pairs given in eq 32 
selects the pair N= 18, 22=5, for which Wi(18, 5) = 
125. Assuming that the computation of S 2 in 

13 This method is expecially useful with respect to computations on Inter- 
national-Business-Machines equipment. 



96 



Journal of Research 



eq 29 involves the same considerations, we may 
therefore propose N=IS, R = 5 as being the 
optimal values to use in getting G(x, z) by eq 33 
and 34 for one value of x and one value of z. 
The number of multiplications will be 2W X (N, R). 
Getting G(x, z) for 72 values of x and one value 
of z involves no change of N, R. Since sin nx 
[cos nx] is an odd [even] function of x, the number 
of multiplications in getting G(x, z) for all x will 
be 72W\(N, R). However, getting G(x, z) for 
several values of z changes the analysis, because 
the functions o- ( £? (x), once computed, serve for each 
new z without change. To get Si for 13 values 
of z and one value of x, for example, will require, 
in addition to the multiplications in eq 41, only 
12iV multiplications from eq 33 and 12R from 
eq 34. The total number of multiplications will 
then be 

W 1Z (N, R) = -66 + l9N+4'3R-NR= 
751-(19-i2)(43-2V). 

The minimum of Wiz(N, R) is 361, and occurs 
for iV=13, R=Q. The optimal choice of N, R 
has changed, though not greatly. Since we 
expect to use 13 values of z, we adopt the values 
JV=13, R--6. 

4. Summary of the Computation Method 

With the above choice of N and R, the compu- 
tation of Si may proceed as follows: 
(a) Compute 



sin nx. 



si =§6 cos C) 9i 

(b) For r=0, 1, 2, compute 



sin nx 

n2r+l > 



where <T2r(x) is computed from section VI. 
(c) Compute 



*£ 3) (x)=-f: s 



(42) 



~4 n' 



^ 13) (*)=S 



sin nx 



i4 n v 



« = 14 n 



(43) 



(d) Compute 

5 z 2r 

^2 — 2-1 (<) T \ | °2r \ X J' 
r = () K" 1 ) ' 

(e) Compute 2i=2){+2j. 

The number of multiplications involved in get- 
ting Si for one x-value and for one .:- value is: 
(a) 13; (b) 39; (c) 69; (d) 6; (e) 0. When getting 
Si for one x-value and 13 2-values, one adds 156 
multiplications to (a) and 72 to (d). The total 
for 13 2-values is 355 multiplications per x-value. 
(The slight discrepancy with the number 361 in 
subsection 3 is due to the rough estimate previously 
made for step (c).) For all ^-values (essentially 
36), one gets a total of 12,780 multiplications to 
get Si. 

To get S 2 in eq 29, one follows analogous steps 
involving o- 2r+ i (x), <r£!%i(x) t etc. There will be 
approximately 12,750 more multiplications, mak- 
ing a total of about 25,500 multiplications to get 
G(x, z) for the 72 x-values and 11 2-values. 

The total truncation error in getting Sj is bounded 
by 2X10 -4 . This is divided into four trunca- 
tion errors of 5X10 -5 , one for each of the three 
steps in (c), and one for the terms left out of (e). 
The truncation error for S 2 is also bounded by 
2 X 10" 4 , making a total truncation error of 4 X 10~ 4 . 
The final round-off of the final answer to three 
decimal places may introduce an error of 5X10 -4 . 
The third source of error is the accumulation of 
round-off s from adding five-decimal-place terms. 
Each term is accurate to 5 X 10" 6 ; with an assumed 
rectangular distribution these terms have a dis- 
persion near 3X10" 6 . Each value of G(x, z) is ob- 
tained from the addition of about 270 such terms. 
The dispersion a of the sum is therefore about 
V270X5X10" 6 , or about 8X10" 5 . One may ex- 
pect the accumulated error to exceed 2.5 a=2X 
10~ 4 in only 1.3 percent of the cases. The sum 
of the three errors is effectively bounded by 
11X10" 4 , or slightly more than 0.001. 

VI. The Polynomials {^(x)} 

In section V we made use of certain functions 
a k (x) defined as follows: 



0"2 



r(*) = (-l) r S 



sin nx 



r~ »" 



(r=0, 1,2,...); 



«T 2r+I (x) = (-l)'+ 1 S 



cos nx 



(r=0,l, 2, ...), 



(44) 



Solution of the Telegrapher's Equation 



97 



The function <r (x) was used in section IV; see eq. 
18. For &>0 the series a k (x) in eq. 44 are abso- 
lutely convergent; hence they represent contin- 
uous functions. Since a 2r (x) is odd and (r 2r+1 (x) is 
even, it is necessary to sum the series (eq. 44) 
only for 0<£<7r. 
As stated in eq. 18, 



7T 1 

<3ro(z) = 2 — 2 X ' (0<^<^)- (45) 



No\ 



; cos nx 



(7, (X) = — 2-1 o~ 

71=1 n 2 



Jo w=ir^ 



Hence 



(Ti(x) = — -w--{-yX — jX 2 > (0<X<7r). 



Similarly, for 0<^c<x, one finds 

1 ,. 



t \ W I T 2 

<T 2 (X) = —^ X + -^X Z - 



12 



cr 2\ X ) = CM\~~To X "1 



90 12 



12 



x° — 



48 



or 4 (a;) = 



90 



36 



a X 3 + 



48 



240 



x°- 



(46) 



Use has been made of the formulas 2T =1 71 2 = 
7r76;S:=i^- 4 = 7r 4 /90. 

The functions <r k (x) are therefore all poly- 
nomials. Their use in improving the convergence 
of Fourier series is pointed out on pp. 84 to 88 
of [10]. Although they may be easily tabulated 
from eq 46, they may also be adapted from exist- 
ing tables because they are essentially Bernoulli 
polynomials. Let {B n (x)\ be the Bernoulli poly- 
nomials given on p. 181 of [4]. 

Lemma. 14 For 0^x<7r, andk = Q, 1, 2, . . . , 



(2tt)* +1 



(0 



**(*)— 2(Jfc+l)! B * +1 



Proof: Define the Bernoulli number B n by the 
relation B n — B n (0) . These are the Bernoulli num- 
bers used on p. 21 of [8]; they are B =l, Bi=— J, 
5 2 =l/6,J5 3 -0,5 4 =-l/30,5 5 =0,5 6 -l/42, 
Davis uses other notations in [4]. Now fix x in 
the interval 0<x<ir. For each k=l, 2, 3, . . . , 



**(*)=* \ <r*-i(&di-" 



. Jct ^\ 1 
sin — 21/ 



2 ZAn** 1 ' 



"After this paper was completed, Professor D. H. Lehmer called the 
author's attention to the statement of this lemma on p. 65 of N. E. Norlund's 
Vorlesungen fiber Differenzenrechnung (Julius Springer, Berlin, 1924). 



Jo '*- i(edf ~2(f+rn 

The last step is by eq 9 on p. 21 of [8], which is 
correct except for sign. Hence, letting x=2irt, 



2(27r)- k - 1 a k (27rt): 



s: 



B, 



2(2ir)-* ff ,_ 1 (2«i|)Ai-^^ I (*=l,2 > 3, . . . ). 

(47) 



Now define 2(2ir) (, ff_,(27rf) = -B /0\(=-l). Use 
eq 47 formally to get 2(2r)~ 1 a (2irt): 

2(2x)- 1 <to(2^) = -§^-^ 1 (=-<+|)- (48) 

Note that eq 48 agrees with eq 45 for a Q (x). Hence 
eq 48 is a correct formula, although it was only 
derived formally. 

We now apply formula (eq 47) repeatedly, 
getting always correct expressions: 

/ n_ 2 /o ,\ B t B\ t B 2 . 

2{Ztt) < T iV* t )--w.2\~V.V.~W 

Of 9 \-3 /o A— ^° ^ Bl ^ ^2 t B 3 . 

Z(2ir) a 2 (M)- - Q -y ^-yy 2!~" 2! f!~ 3l'' 



Hence 



w 1 ^ 2 ^— SflFT^r 



2(t + 1) l(2,r)-'-V2.f) = -g ( ^f^7 
* fk+V 



-sCi 1 )^ 1 "- 



(49) 



But it follows from the top of p. 188 of [4] that 
B k+l (t) =S C^r 1 ) Bf-»-i- (50) 

Comparing eq 49 and 50, we see that 

B k+1 (t) = -2(k + l)\(2ir)- k - 1 a k (27rt). 
Let 2irt=x, and the lemma is proved. 

VII. Solution by a Difference Equation 

Our first approximate solution of the problem 
stated in section II consisted of the approximate 
evaluation of the integral (eq 21) by means of 
numerical integration formulas, using the approxi- 



98 



Journal of Research 



mate values of 0(x, z) tabulated in section VIII 
below. A second approximate solution of the 
problem consists in solving with appropriate 
boundary conditions a difference equation that is 
closely related to the differential equation (eq 3). 
The latter method is considered in detail in [6], 
where proofs may be found; only a summary is 
given in the present section. 

For any positive integer 2N, let h=w/2N; let 
&>0 be arbitrary. A net is formed of all points 
(Xj t) of form (fih, vk) , where /x and v are integers 
satisfying the conditions 

fx+v = (mod 2), \n\<2N, ^>0. (51) 

Where necessary we extend the net and the func- 
tional values periodically in x with period 2w. 
The differential equation (eq 3) is approximated 
by the difference equation, 

v(x-\-h, t+k)—v(x—h, t-\-k) = 

v(x+h, t—k)—v(x—h, t-k)—hkv(x, t). (52) 

The boundary conditions of the difference-equa- 
tion problem are prescribed values of v(x, t) on the 
two rows £=0, t=k. Assume that for t=k, 



2>CM)=o, 



(53) 



where the sum is extended over all points of the 
second row of the net. The boundary conditions 
and eq 52 then determine the value of v(x, t) on the 
row t=2k up to an additive constant. The 
additive constant and hence v(x, 2k) are determined 
uniquely by requiring that eq 53 hold also for 
t = 2k. Continuing row after row, one thus de- 
termines v(x, t) over the whole net. Let the 
function so determined be denoted by v m (x, f); it 
depends on N, on k, and on the initial values pre- 
scribed for the first two rows. The problem of [6] 
is to see whether v iN) (x, t)-^>v(x, t) as iV— > oo . 

Let the initial values v(x, 0) on the first row be 
defined by the relation v(x, 0)=f(x), where j(x) is 
the function of eq 1. Let k be fixed. Then it is 
possible to choose the initial values v(x y k) on the 
second row of the net in such a manner that, as 
Af-^oo } v m (x, t)->v(x,t) for each t of the net and 
for each x that is an abscissa of continuity oij(x). 
If k is allowed to vary with N in such a manner 
that k—>0 as N-^co ) then v {N) (x, t)-^v{x, t) for each 
t>0 and for each x that is an abscissa of conti- 
nuity o{J(x) . In neither case may one, in general, 
expect the convergence to be uniform in x or t. 



The method referred to for choosing the values 
v(x, k) on the second row is not an economical 
one, and in a practical computation one would 
prefer a cheaper though approximate method. 
Two things are shown in [6] about the effects of an 
approximation of the values of v(x, k) : First, they 
may introduce ultimate instability into the solu- 
tion. Even though the solution v(x, t) of eq 3 
be identically zero, it is possible that for fixed 
N and x, 

\imv {N) (x,t) = + oo. 

Second, the approximation docs not prevent con- 
vergence of v {N) (x,t) to v(x,t), provided that the 
error of the approximation of v(x,k) vanishes as 
iV-^oo. One reasonable way of causing the error 
to vanish is to let &— >0. 

These results show that the difference-equation 
method is a feasible method of solving the 
problem of this paper. 

VIII. Table of the Green's Function 

In this section is tabulated the Green's function 
G(x, z), as computed in the Computation Unit of 
the Institute for Numerical Analysis. The value of 
the time parameter z corresponding to h hours at 
latitude <j> is 

2=0.52502 h cos 2 <j>. (51) 

(Except for the last digit of the constant, formula 
(eq 51) can be verified from the introduction to 
section V.) Meteorological considerations sug- 
gested that h should be chosen in convenient mul- 
tiples of 12 hours, and that </> should be 35°, 45°, or 
55°. The latitude 32°19' resulted from a numer- 
ical error by the author. A limited number of 
pairs of values of h and <j> were selected for the 
computation; these pairs arc shown in table 1, 
together with the corresponding values of z 
determined from eq 51. 

For each of the 13 values of z given in table 1 
(and for z=0) and for x=— x(7r/36)ir, the Green's 
function G(x, z) is presented in table 2 to 5 deci- 
mal places. Since 6? (#, z) has a discontinuity of 
the first kind at x=0, the values c7( — 0, z) } G(0, z), 
and 67(+0, z) arc all given. In every case, 
67(0, 3)=i[G(-0, z) + G(0, z)] and G(+0, z) — 
G( — 0, z) = 7r. The computational procedure fol- 
lowed that of section V in general outline, with 
certain deviations. It was decided to use N=18, 
i?=6. The auxiliary functions o- ( £? (x) and 



Solution of the Telegrapher's Equation 



99 



o"2r+\ 0*0 were computed from formulas like eq 36; 
formulas like eq 38 were not used. This necessi- 
tated carrying considerably more than 10 digits, 
and so the polynomials <r k (x) were first computed 
to 17 decimal places. Choice of formula (eq 36) 
was based on the value of getting these tables of 
Bernoulli polynomials as a byproduct of general 
interest. 

To be definite/ 5 let us write G{x, z) = G 1 (x, 2) + 
G 2 (x, z) , where 

G 1 (x i z)=T,lsm(nx+^\ (52) 

and 

G 2 (x,z) = ^2 -sin(nx-\ — )• 

Let 

(b k /dz k )G 2 (x,z) = G ifc) (x,z),tork=0, 1,2, .... 
Then 

G {k) (x, 0)=S -tt sin (nx+Ic %)=g k (x) —h k (x), 
n =iQ n ^ \ zy 

where 

0*0*0 =S ^ffi sin (nx+k | j, 

By section VI, the functions ^(z) are polynomials. 
They were generated on an International-Busi- 
ness-Machines tabulator to 17 decimal places. 
The values of h k (x) were computed and subtracted 
from g k (x) to yield G (k) (x, 0). For any z, 



G 2 (x,z)=Jly ] GM(x,0)+R k (x ) z), 

where there exists a z x (0<2!<2) such that 
z h 



(53) 



|ff*(*,2)l<- 



-,K+1 



^ (K+ "(X, g,)- 



(K+l)! 

If 0<z<36,2 n /ll!<.33X10 10 . And \G iu \x, Si)|< 
S«" 12 <1-5X10- 15 . Hence, for 0<z< 36, 

|fl„(x,g)l< 0.5X10- 5 . 

15 This description of the computational procedure was furnished by 
Gertrude Blanch of the Computation Unit, Institute for Numerical Analysis 



Once the values of G (k) (x, 0) were obtained it was 
possible to generate, very easily, the function 
G(x,z) for any values of z in the range 0<2<36. 
To summarize, 

(a) Gi(x, z) was computed from eq (52). 

(b) The functions G ik) (x, 0) were computed 
for £==0,1,2, . . . , 10. 

(c) G 2 (x } z) was obtained from eq (53). 

(d) Gi(x, z) and G 2 (x, z) were added, to yield 
G(x, z). 

The subsidiary computations in (a) and (b) were 
carried to nine decimal places, those in (c) to at 
least seven decimal places in the partial products. 
Table 2 is believed to be accurate to ±0.00002 
for all x and all z. The cosine component of 
G(x, z) was given a final check by use of the fol- 
lowing formula: 



35 oo n - 

S G{k*lW J z)=^±-sm~= V {z)- 

fe = -36 v = l v 



72/ 



The check showed a deviation between the sum 
and p(z), which was never greater than 0.00005. 
The sine component of G (x, z) was given a partial 
check by the formula 



35 k 

XI sin -pr- G(Jct/36,z) =2 cos 7-5+ 

k = -3Q Z 15 



2±T- 1 



L>+i 



cos 



(: 



8/18 \ 1_ 

4v+l) 4?— 1 



cos 



am 



Comparable agreement was found. Finally, the 
table differences with respect to x are very 
satisfactory. 

Table 1. Values of the time parameter, z, corresponding 
to hours, h, at latitude, <f> 







4> 




h 




















32°19' 


35° 


45° 


55° 


Hours 










12 






3. 150 




24 


9.000 


8.455 


6.300 


4.145 


48 


18. 000 


16. 909 


12.600 


8.291 


72 


27. 000 




18. 900 




96 


36. 000 




25. 200 





100 



Journal of Research 



Table 2. Table of Green's Function G(x, z). 



Table 2. Table of Green's Function G(x, z). 











2 








X 


















0.00000 


3.15012 


4.14543 


6.30024 


8.29086 


8.45505 


9.00000 


Degrees 














-180 


-0.00000 


0. 32695 


1.08321 


-0. 16762 


-1.34038 


-1.26672 


-0. 85722 


-175 


-. 04363 


. 39530 


1.08444 


-.30315 


-1.29132 


-1.19935 


-.74063 


-170 


-.08727 


. 45756 


1.07647 


-. 43082 


-1. 22533 


-1.11601 


-.61286 


-165 


-. 13090 


. 51370 


1.05993 


—.54955 


-1.14399 


-1.01848 


-.47630 


-1G0 


-. 17453 


. 56372 


1.03552 


-.65844 


-1.04899 


-0. 90869 


-.33337 


-155 


-.21817 


. 60768 


1.00392 


-.75679 


-0. 94217 


-.78862 


-.18646 


-150 


-.26180 


. 64565 


0. 96581 


-. 84405 


-.82542 


-. 66032 


-.03791 


-145 


-.30543 


. 67775 


. 92189 


-. 91985 


-. 70067 


-. 52583 


. 11005 


-140 


-. 34907 


. 70413 


. 87284 


-.98398 


-. 56985 


-.38717 


. 25533 


-135 


-. 39270 


. 72496 


. 81933 


-1.03636 


-. 43487 


-. 24630 


. 39595 


-130 


-. 43633 


. 74042 


. 76202 


-1.07704 


-. 29757 


-. 10512 


. 53015 


-125 


-. 47997 


. 75072 


. 70154 


-1.10620 


-. 15975 


. 03460 


. 65632 


-120 


-. 52360 


. 75611 


. 63851 


-1.12412 


-.02308 


.17118 


. 77306 


-115 


-. 56723 


. 75679 


. 57351 


-1.13116 


. 11085 


. 30309 


. 87916 


-110 


-. 61087 


. 75304 


. 50712 


-1.12779 


. 24058 


. 42893 


. 97363 


-105 


-. 65450 


. 74510 


. 43985 


-1.11454 


. 36479 


. 54746 


1. 05568 


-100 


-. 69813 


. 73324 


. 37221 


-1.09199 


. 48229 


. 65759 


1. 12472 


-95 


-. 74176 


.71771 


. 30468 


-1.06078 


. 59203 


. 75838 


1. 18034 


-90 


-. 78540 


. 69880 


. 23769 


-1.02 ICO 


. 69310 


. 84906 


1. 22233 


-85 


-. 82903 


. 67677 


. 17165 


-0.97516 


. 78475 


. 92902 


1. 25066 


-80 


-. 87266 


. 65189 


. 10695 


-.92219 


. 86637 


. 99779 


1.26545 


-75 


-. 91630 


. 62442 


. 04392 


-.86344 


. 93749 


1. 05505 


1. 26700 


-70 


-0. 95993 


. 59462 


-.01712 


-.79967 


. 99777 


1. 10063 


1. 25573 


-65 


-1.O0356 


. 56277 


-.07590 


-.73162 


1. 04701 


1. 13450 


1.23218 


-60 


-1.04720 


. 52910 


-.13215 


-.66004 


1. 08513 


1.15674 


1.19701 


-55 


-1.09083 


. 49386 


-.18566 


-.58566 


1.11218 


1. 16755 


1. 15098 


-50 


-1.13446 


. 45730 


-.23624 


-. 50920 


1.12832 


1. 16724 


1.09494 


-45 


-1.17810 


. 41965 


-.28373 


-.43136 


1. 13378 


1.15623 


1. 02978 


-40 


-1.22173 


.38113 


-.32798 


-.35278 


1. 12892 


1. 13502 


0. 95646 


-35 


-1.26536 


. 34197 


-.36889 


-.27411 


1. 11416 


1. 10417 


. 87600 


-30 


-1.30900 


. 30236 


-.40638 


-.19596 


1. 09002 


1.06431 


. 78941 


-25 


-1.35263 


. 26250 


-.44037 


-. 11887 


1. 05705 


1. 01616 


. 69774 


-20 


-1.39626 


. 22259 


-.47084 


-.04338 


1. 01588 


0. 96044 


. 60203 


-15 


-1.43990 


. 18280 


-.49776 


.03002 


0. 96717 


. 89794 


. 50333 


-10 


-1.48353 


. 14330 


-.52114 


. 10089 


.91164 


. 82944 


.40264 


-5 


-1.52716 


. 10426 


-.54100 


. 16881 


. 85000 


. 75578 


.30096 


-0 


-1.57080 


.06582 


-.55736 


. 23344 


. 78302 


. 67778 


. 19926 





0. 00000 


1. 63662 


1. 01344 


1.80424 


2. 35382 


2. 24858 


1. 77006 


+0 


1. 57080 


3. 20741 


2. 58423 


3. 37503 


3. 92461 


3. 81938 


3. 34085 


5 


1. 52716 


2. 36366 


1. 53355 


1.93219 


1. 95959 


1. 81396 


1. 21684 


10 


1. 48353 


1. 62904 


0. 66827 


0. 87990 


0. 63440 


0. 47195 


-0. 16296 


15 


1.43990 


0. 99385 


-.03330 


. 14729 


-. 20075 


-. 36465 


-. 98574 


20 


1. 39626 


. 44908 


-.59093 


-.32689 


-. 66889 


-. 82505 


-1.40257 


25 


1. 35263 


-.01369 


-1.02264 


-. 59533 


-. 87018 


-1. 01399 


-1. 53397 


30 


1.30900 


-. 40226 


-1.34473 


-. 70310 


-. 88521 


-1.01530 


-1. 47484 


35 


1. 26536 


-. 72390 


-1.57202 


-.68848 


-. 77804 


-0. 89523 


-1. 29869 


40 


1. 22173 


-.98534 


-1.71788 


-. 58382 


-. 59873 


-. 70520 


-1. 06130 


45 


1. 17810 


-1.19282 


-1.79437 


-.41615 


-. 38563 


-. 48429 


-0. 80387 


50 


1. 13446 


-1.35212 


-1.81235 


-.20791 


-. 16734 


-. 26137 


-. 55574 


55 


1.09083 


-1.46858 


-1.78153 


. 02253 


. 03555 


-.05690 


-.33671 


60 


1.04720 


-1.54714 


-1.71063 


. 26034 


. 20903 


. 11545 


-.15898 


65 


1. 00356 


-1.59233 


-1.60740 


. 49376 


. 34437 


. 24746 


-.02884 


70 


0. 95993 


-1.60833 


-1.47871 


.71367 


. 43706 


. 33521 


. 05197 


75 


. 91630 


-1.59898 


-1.33066 


.91321 


. 48585 


. 37809 


. 08507 


80 


.87266 


-1.56780 


-1.16858 


1. 08748 


. 49200 


. 37799 


. 07448 











z 








X 


0. 00000 


3. 15012 


4. 14543 


6. 30024 


8. 29086 


8. 45505 


9.00000 


Degrees 
















85 


. 82903 


-1.51800 


-0. 99717 


1. 23323 


. 45862 


.33860 


.02585 


90 


. 78540 


-1.45251 


-. 82049 


1.34855 


.39011 


. 26486 


-.05417 


95 


.74176 


-1.37403 


-. 64207 


1.43272 


. 29171 


.16247 


-. 15846 


100 


. 69813 


-1.28496 


-. 46492 


1.48592 


. 16915 


.03753 


-.27984 


105 


. 65450 


-1.18751 


-.29158 


1. 50910 


. 02832 


-. 10376 


-.41132 


110 


. 61087 


-1.08367 


-. 12420 


1. 50380 


-. 12493 


-. 25537 


-. 54637 


115 


. 56723 


-0. 97523 


. 03545 


1. 47204 


-. 28498 


-.41155 


-.67907 


120 


. 52360 


-.86377 


. 18594 


1.41614 


-. 44662 


-. 56703 


-.80424 


125 


. 47997 


-.75074 


.32616 


1.33870 


-.60507 


-. 71708 


-.91745 


130 


. 43633 


-.63740 


. 45522 


1. 24244 


-.75611 


-. 85757 


-1.01507 


135 


. 39270 


-.52487 


. 57252 


1. 13016 


-.89609 


-.98499 


-1.09431 


140 


. 34907 


-.41413 


. 67764 


1.00467 


-1.02197 


-1.09649 


-1.15312 


145 


. 30543 


-.30603 


. 77037 


0. 86874 


-1.13129 


-1.18985 


-1.19019 


150 


. 26180 


-.20133 


. 85064 


. 72505 


-1.22217 


-1.26346 


-1.20491 


155 


.21817 


-.10064 


. 91855 


. 57617 


-1.29329 


-1.31627 


-1.19725 


100 


. 17453 


-.00450 


. 97429 


. 42452 


-1.34386 


-1.34779 


-1.16777 


165 


. 13090 


. 08665 


1.01819 


. 27234 


-1.37355 


-1.35800 


-1.11746 


170 


.08727 


. 17245 


1. 05065 


.12170 


-1.38248 


-1.34730 


-1.04773 


175 


. 04363 


. 25262 


1.07214 


-.02551 


-1.37114 


-1.31650 


-0. 96032 


180 


. 00000 


. 32695 


1. 08321 


-. 16762 


-1.34038 


-1.26672 


-.85722 




12.60048 


16.91010 


18.00000 


18.90072 


25.20096 


27.00000 


36.00000 


-180 


0. 21759 


1. 38333 


0.88710 


-0. 22251 


-0. 23708 


-0. 52214 


0. 75323 


-175 


. 20848 


1.28719 


. 68046 


-.44845 


-. 13949 


-.34513 


. 66966 


-170 


.18520 


1.17912 


.47714 


-. 65407 


-.04531 


-.19161 


. 59996 


-165 


.14768 


1.06395 


. 28213 


-. 83538 


. 03742 


-.06711 


. 55216 


-160 


.09639 


0.94615 


. 09946 


-.98956 


. 10193 


.02510 


. 53133 


-155 


. 03224 


. 82961 


-.0674'; 


-1.11498 


. 14294 


.08392 


. 53944 


-150 


-.04345 


.71765 


-.21599 


-1. 21108 


. 15684 


.11027 


. 57545 


-145 


-.12901 


. 61292 


-. 34473 


-1.27829 


.14173 


. 10689 


. 63557 


-140 


-. 22253 


. 51745 


-. 45323 


-1.31787 


. 09736 


. 07795 


. 71381 


-135 


-.32192 


. 43256 


-.54152 


-1.33178 


. 02503 


. 02877 


. 80257 


-130 


-.42494 


. 35899 


-.61014 


-1.32256 


-.07262 


-. 03461 


. 89326 


-125 


-. 52932 


. 29691 


-. 66035 


-1.29314 


-. 19180 


-.10578 


. 97703 


-120 


-. 63276 


. 24596 


-.69412 


-1.24669 


-. 32784 


-. 17834 


1. 04492 


-115 


-.73306 


. 20536 


-. 71352 


-1.18651 


-.47538 


-. 24602 


1.09064 


-110 


-.82808 


. 17397 


-.72062 


-1.11592 


-. 62883 


-.30340 


1. 10661 


-105 


-.91587 


. 15038 


-.71767 


-1.03810 


-.78241 


-.34564 


1. 08871 


-100 


-. 99463 


. 13297 


-. 70728 


-0. 95602 


-. 93051 


-. 36894 


1. 03429 


-95 


-1.06278 


. 12001 


-.69185 


-. 87240 


-1.06785 


-.37053 


0. 94265 


-90 


-1. 11899 


. 10975 


-. 67337 


-. 78959 


-1.18968 


-.34876 


.81504 


-85 


-1. 16216 


. 10043 


-. 65362 


-. 70959 


-1.29191 


-. 30309 


.65453 


-80 


-1. 19145 


. 09041 


-. 63445 


-.63398 


-1.37124 


-.23406 


. 46575 


-75 


-1.20628 


. 07817 


-. 61745 


-. 56396 


-1.42518 


-. 14319 


.25459 


-70 


-1.20633 


. 06238 


-. 60358 


-. 50029 


-1.45213 


-.03286 


. 02788 


-65 


-1.19154 


. 04192 


-. 59343 


-. 44339 


-1.45134 


.09382 


-. 20697 


-60 


-1. 16210 


. 01592 


-. 58753 


-. 39331 


-1.42289 


.23321 


-. 44192 


-55 


-1. 11840 


-.01623 


-. 58625 


-.34978 


-1.36773 


.38109 


-.67081 


-50 


-1.06108 


-. 05488 


-. 58940 


-.31226 


-1.28738 


.53326 


-.88534 


-45 


-0. 99094 


-.10011 


-. 59636 


-. 28000 


-1.18407 


. 68527 


-1.07970 


-40 


-.90897 


- 15176 


-. 60651 


-.25205 


-1.06049 


. 83281 


-1.24859 


-35 


-.81632 


-. 20939 


-. 61926 


-.22736 


-0.91976 


. 97176 


-1.38784 



Solution of the Telegrapher's Equation 



101 



Table 2. Table of Green's Function G(x, z). 





2 


X 


12.60048 


16.91010 


18.00000 


18.90072 


25.20096 


27.00000 


36.00000 


Degrees 
















-30 


-. 71423 


-.27239 


- 63367 


- 20477 


-. 76527 


1. 09836 


-1. 49453 


-25 


-.60408 


-. 33990 


-.64843 


-.18315 


-.60056 


1. 20927 


-1. 56697 


-20 


-. 48731 


-. 41091 


-.66232 


-. 16133 


-.42926 


1. 30167 


-1. 60479 


-15 


-.36540 


-. 48428 


-.67437 


-. 13823 


-.25493 


1. 37335 


-1. 60877 


-10 


-. 23989 


-. 55874 


-.68350 


-.112S6 


-. 08102 


1. 42266 


-1. 58O80 


-5 


-.11230 


-.63295 


-.68839 


-.08436 


. 08924 


1. 44862 


-1. 52370 


-0 


.01585 


-. 70554 


-.68784 


-.05201 


. 25293 


1. 45084 


-1.44108 





1. 58665 


. 86526 


. 88296 


1. 51879 


1.82373 


3. 02164 


0. 12972 


+0 


3. 15744 


2. 43605 


2. 45375 


3. 08958 


3. 39452 


4. 59243 


1. 70O52 


5 


0. 67142 


-0. 81525 


-0. 84342 


-0. 27249 


-0. 37260 


0. 54092 


-2. 54939 


10 


-. 51209 


-2. 00417 


-1.87971 


-1. 21362 


-.63490 


. 27468 


-1. 76069 


15 


-. 85078 


-2. 07925 


-1. 75697 


-0. 97035 


. 26636 


1.10072 


-0. 49519 


20 


-. 68020 


-1.65134 


-1. 16076 


-. 29000 


1. 20809 


1.82328 


-. 00356 


25 


-.24015 


-1. 09098 


-0. 48829 


. 4121.3 


1. 76793 


2. 06387 


-. 20922 


30 


. 30379 


-0. 60220 


. 05701 


. 93791 


1.89206 


1. 84177 


-. 68549 


35 


. 84267 


-.27771 


. 39671 


1. 22420 


1.69102 


1. 32871 


-1.06815 


40 


1. 31002 


-.13946 


. 52770 


1.28534 


1.32178 


0. 71692 


-1. 17421 


45 


1. 67048 


-.16770 


. 48736 


1. 17420 


0. 92635 


. 15777 


-0. 98652 


50 


1. 91086 


-.32146 


. 33030 


0. 95687 


. 60563 


-. 25779 


-. 59209 


55 


2. 03312 


-. 55228 


. 11399 


.69716 


.41414 


-. 49626 


-. 11820 


60 


2. 04891 


-.81299 


-.11013 


. 44820 


. 36577 


-. 56829 


. 31487 


65 


1. 97550 


-1. 06293 


-.30155 


. 24868 


. 44447 


-. 51226 


. 62341 


70 


1. 83275 


-1.27046 


-. 43250 


. 12229 


. 61561 


-. 37956 


. 76541 


75 


1. 64092 


-1. 41383 


-.48723 


. 07901 


. 83661 


-. 22221 


. 74192 


80 


1.41914 


-1. 48076 


-.46049 


. 11749 


1. 06504 


-. 08486 


. 58453 


85 


1.18449 


-1. 46731 


-.35588 


. 22780 


1. 26435 


. 00000 


. 34287 


90 


0. 95137 


-1.37647 


-. 18386 


. 39425 


1. 40727 


. 01370 


. 07206 


95 


. 73134 


-1.21644 


. 04076 


. 59799 


1.47711 


-.04915 


-. 17737 


100 


. 53301 


-0. 99910 


. 30108 


.81915 


1. 46763 


-. 18262 


-.36632 


105 


. 36223 


-. 73849 


. 57953 


1. 03865 


1. 38174 


-. 37234 


-.47053 


110 


. 22235 


-.44954 


. 85891 


1. 23947 


1. 22959 


-. 59864 


-.48116 


115 


. 11450 


-. 14704 


1.12358 


1. 40751 


1. 02640 


-.83951 


-. 40310 


120 


. 03798 


. 15526 


1.36058 


1. 53214 


0. 79018 


-1.07324 


-.25141 


125 


-.00939 


. 44516 


1. 55988 


1. 60630 


. 53965 


-1.28065 


-. 04972 


130 


-. 03091 


. 71247 


1.71412 


1. 62645 


. 29276 


-1. 44613 


. 17798 


135 


-. 03066 


. 94918 


1.81868 


1. 59230 


. 06507 


-1.55895 


. 40643 


140 


-.01322 


1. 14954 


1.87196 


1 . 50636 


-. 13095 


-1. 61332 


. 61405 


145 


. 01665 


1.31003 


1.87498 


1. 37347 


-.28656 


-1. 60827 


. 78435 


150 


. 05419 


1.42912 


1.83068 


1.20024 


-.39683 


-1. 54718 


. 90692 


155 


. 09488 


1. 50714 


1.74337 


0. 99449 


-. 46051 


-1. 43694 


. 97765 


160 


. 13453 


1. 54593 


1.61879 


. 76479 


-.47971 


-1.28710 


. 99822 


165 


. 16945 


1. 54859 


1 . 46378 


. 51991 


-.45930 


-1. 10889 


. 97514 


170 


. 19650 


1. 51918 


1 . 28553 


. 26848 


-.40628 


-0. 91426 


. 91844 


175 


.21318 


1. 46239 


1.09103 


. 01857 


-.32913 


-. 71502 


. 84021 


180 


.21759 


1.38333 


0. 88710 


-.22251 


-.23708 


-. 52214 


. 75323 



IX. References 

[I] J. Charney, A. Eliassen, G. Hunt, Report on a program 

for numerical weather predicting, paper presented 
at a meeting of the American Meteorological Society, 
New York, N. Y., January 28, 1949. 

[2] J. Charney, manuscript in preparation for Journal 
of Meteorology on the work reported in [1]. 

[3] R. Courant, Differential and integral calculus (trans- 
lated by E. J. McShane) 2 (Blackie and Sons, 
London and Glasgow, 1936). 

[4] Harold T. Davis, Tables of the higher mathematical 
functions, 2 (Principia Press, Bloomington, Ind., 
1935). 

[5] G. E. Forsythe, J. Meteorology 4, 67 (1947). 

[6] G. E. Forsythe, Manuscript in preparation on the 
approximate solution by difference equations of the 
telegrapher's equation with boundary values on only 
one characteristic. 

[7] G. E. Forsythe, A solution of the telegrapher's equation 
with initial conditions on only one characteristic. 
Preliminary Report. Paper presented at a meeting 
of American Mathematical Society, Columbus, Ohio, 
December 30, 1948. 

[8] Erwin Madelung, Die Mathematischen Hilfsmittel des 
Physikers, American ed. (Dover Publications, New 
York, N. Y., 1943). 

[9] C. G. Rossby and collaborators, J. Marine Res. 2, 38 

(1939). 

[10] J. D. Tamarkin and Willy Feller, Partial differential 
equations (mimeographed lecture notes, Brown 
University, Providence, R. I., 1941). 

[II] A. G. Webster, Partial differential equations of 

mathematical physics, 2d corrected ed. (Hafner 
Publishing Co., New York, N. Y., 1947). 

[12] A. Zygmund, Trigonometrical Series (Monografje 
Matematyczne, Warsaw-Lwow, 1935). 



Los Angeles, March 8, 1949. 



102 



Journal of Research